2.2.1 Box cones🔗ℹ

The box cone is {(t, r) : t·l ≤ r ≤ t·u} with a leading scale variable t. To impose plain bounds l ≤ y ≤ u, arrange A and b so the first box row’s slack is a constant 1 (a row of zeros in A with the matching b entry equal to 1) and the remaining rows carry y. make-cone takes the bounds with #:box-lower and #:box-upper; their length is the number of box-constrained entries, and SCS’s bsize is that length plus one. Model predictive control is a worked example.

2.2.2 Semidefinite cones🔗ℹ

For a k×k symmetric matrix, the PSD cone slack is the lower triangle stacked column-wise with the off-diagonal entries scaled by √2; the block length is k(k+1)/2. So a 2×2 matrix [[a b] [b c]] is represented by (a, √2·b, c). #:psd takes the list of k values. See the cones reference and Semidefinite program.

2.4.1 Laying out A in cone-ordered row-blocks🔗ℹ

The rows of A (and the entries of b and s) must appear in the Cones order: zero rows, then positive, box, second-order, semidefinite, exponential, and power. Build A as a stack of row-blocks, one per primitive cone, in that order, and make make-cone’s keywords describe the same blocks. A slack lands in a given cone exactly when s = b − Ax restricted to that block satisfies the membership condition; a common idiom is an A = −I block with b = 0, which sets s = x on those rows so the variables themselves are constrained (Second-order cone, Exponential cone, Power cone).

2.4.2 Bounds and the absolute-value trick🔗ℹ

A two-sided bound l ≤ aᵀx ≤ u is two positive-orthant rows, aᵀx ≤ u and −aᵀx ≤ −l. The absolute-value trick models |aᵀx|: introduce an auxiliary t, add the two rows aᵀx − t ≤ 0 and −aᵀx − t ≤ 0 (so t ≥ |aᵀx|), and use t in the objective. This is how L1 penalties become cone programs in Lasso along a regularization path and Elastic net regression.

2.4.3 Norms via the second-order cone🔗ℹ

A Euclidean-norm bound ‖u‖₂ ≤ t is one second-order cone block over (t, u) (Second-order cone). To minimize a norm, bound it by a fresh variable t with ‖u‖₂ ≤ t and minimize t.

2.4.4 The epigraph trick for nonlinear objectives🔗ℹ

To minimize a convex function f(x) that is not already linear or quadratic, minimize a new variable t subject to the epigraph trick constraint f(x) ≤ t, then express that constraint with a cone. For x log x the epigraph is exponential-cone membership, which is how Maximum entropy maximizes entropy.

2.4.5 Box-cone layout🔗ℹ

The box cone carries a leading scale variable. To impose plain bounds l ≤ y ≤ u, pin that scale to 1 with a constant row (zeros in A, a 1 in b) and put y on the remaining rows; see Box cones and the worked Model predictive control.

2.4.6 Symmetric matrices and the svec scaling🔗ℹ

A semidefinite constraint X ⪰ 0 is one #:psd block whose slack is the √2-scaled lower-triangular vectorization of X; see Semidefinite cones and Semidefinite program.

2.6.1 Direct vs. indirect solver🔗ℹ

By default solve uses the direct linear-system solver. Pass #:indirect? #t to use the indirect (conjugate-gradient) solver, which can scale better on very large sparse problems.

2.6.2 Warm starting and re-solving🔗ℹ

When a sequence of problems differs only in b or c, build a solver once with make-solver and re-solve it instead of rebuilding from scratch. solver-solve! returns an scs-result; it cold-starts the first time and warm-starts (reusing the previous iterate) on every call after that. solver-update! swaps in a new b and/or c (the matrices and cone are fixed):

(define s (make-solver #:A A #:b b #:c c #:cone cone))

(solver-solve! s)              ; cold start

(solver-update! s #:b b2)      ; only b changes

(solver-solve! s)              ; warm-started re-solve

solve is just make-solver followed by a single solver-solve!. Warm starting and re-solving, Lasso along a regularization path, and Model predictive control show the pattern. For full manual control over the workspace and solution buffers, the scs/foreign layer exposes the underlying scs-init / scs-update / scs-solve.

2.6.3 Troubleshooting and status🔗ℹ

When solved? is false, the exit flag says why, and the result still carries useful information:

• Infeasible (flag -2): no x satisfies the constraints. SCS returns a certificate of infeasibility in scs-result-y (a dual ray); scs-result-x is not meaningful.

• Unbounded (flag -1): the objective can be driven to −∞ within the constraints. The certificate is a primal ray in scs-result-x.

• Inaccurate (flags 2, -6, -7): SCS hit its iteration or time limit with the residuals only partially converged. The iterate is usually close; loosen #:eps-abs / #:eps-rel to accept it, or raise #:max-iters / #:time-limit-secs to push further.

• Indeterminate / failed (flags -3 through -5): the problem could not be classified, or setup failed.

A few common modeling mistakes produce a spurious infeasible or unbounded result rather than an error:

• Cone-order mismatch. The rows of A must follow the Cones order and match make-cone’s keyword counts exactly. A miscounted block silently re-interprets later rows.

• A missing box scale row. The box cone needs its leading scale entry pinned (see Box cones); omitting it changes the feasible set.

• Wrong PSD scaling. Off-diagonal entries of a semidefinite block must be √2-scaled (see Semidefinite cones).

To see SCS’s own iteration log while diagnosing, pass (make-settings #:verbose? #t).